Multiple Model Predictive Functional Control for Marine Diesel Engine

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 3252653, 20 pages https://doi.org/10.1155/2018/3252653

Research Article Multiple Model Predictive Functional Control for Marine Diesel Engine Runzhi Wang

, Xuemin Li

, Yufei Liu

, Wenjie Fu , Shuang Liu , and Xiuzhen Ma

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Xuemin Li; [email protected] Received 11 December 2017; Accepted 28 March 2018; Published 14 May 2018 Academic Editor: Xinkai Chen Copyright © 2018 Runzhi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A novel control scheme based on multiple model predictive functional control (MMPFC) is proposed to solve the cumbersome and time-consuming parameters tuning of the speed controller for a marine diesel engine. It combines the MMPFC with traditional PID algorithm. In each local linearization, a first-order plus time delay (FOPTD) model is adopted to be the approximate submodel. To overcome the model mismatches under the load disturbance conditions, we introduce a method to estimate the open-loop gain of the speed control model, by which the predictive multimodels are modified online. Thus, the adaptation and robustness of the proposed controller can be improved. A cycle-detailed hybrid nonlinear engine model rather than a common used mean value engine model (MVEM) is developed to evaluate the control performance. In such model, the marine engine is treated as a whole system, and the discreteness in torque generation, the working imbalance among different cylinders, and the cycle delays are considered. As a result, more reliable and practical validation can be achieved. Finally, numerical simulation of both steady and dynamic performances of the proposed controller is carried out based on the aforementioned engine model. A conventional welltuned PID with integral windup scheme is adopted to make a comparison. The results emphasize that the proposed controller is with stable and adaptive ability but without needing complex and tough parameters regulation. Moreover, it has excellent disturbance rejection ability by modifying the predictive multimodels online.

1. Introduction Diesel engines are the most pervasive and favored prime movers or power sources in the domain of ship for their superior efficiency [1–3]. Speed control is the crucial task for the diesel engines, because the engine performance and service life of the engine and ship rely very much on the speed adjusting [4]. Especially for the diesel engines serving as the marine main engine for propulsion, it is necessary to govern the engine speed during the whole operating processes. Otherwise, the oscillation of engine speed leads to the engine being unable to operate normally [1]. Moreover, the overspeed caused by poor speed governing performance will bring irreversible damage in the marine main engine [5]. It should be pointed out that large speed fluctuation of the engine may cause heavy vibration and strong impact forces, inducing damaging vibration in the related structures leading to premature failure of the transmission system [6]. Hence, the crucial control targets should include (1) tracking the

speed set-point quickly and keeping the possible smallest deviations from the desired speed and (2) recovering fast from the external disturbances. According to the practical needs, speed control strategies for the marine diesel engines have drawn considerable attention during the last decades. However, the speed control algorithm design for the marine diesel engine remains to be a tough mission due to its inherent high nonlinearity. As for the engine speed, it is a function of various aspects, such as the ambient temperature and humidity, fuel injection timing, and compression and combustion processes. And some of these parameters also rely on the transient engine speed to a larger extent at the same time [7]. Furthermore, speed and load of the marine main engine are strongly affected by numerous other external aspects, such as the weather in the sea and the surface condition of the sea [2, 8]. Attempts have been made on the basis of variable control method, such as conventional PID method [9], sliding mode control (SMC) [2], 𝐻∞ control [10],

2 active disturbance rejection control (ADRC) [11], fuzzy control [5], and model predictive control (MPC) [12]. Although remarkable progress has been acquired, there are still some drawbacks. Specifically, 𝐻∞ control and MPC techniques not only rely on exact mathematical model, but also require high-performance processor to execute the complex matrix computation. As shown in [12], the MPC method was carried out with the help of two 900-MHz IMB PPC processors, which actually cannot be accepted commercially. Despite the fact that there exists computationally efficient MPC algorithm with online linearization and quadratic optimization (e.g., [13–15]), where the online computational requirement is acceptable, such computational simplicity is generally realized by sacrificing the control performance as the linear approximation is invalid when the system state and input deviate far away from where they are linearized. Meanwhile, all these advanced methods still suffer from tiring tuning process [9]. As a result, most of the commercial controllers for marine main engine are based on PID or PI [3, 4, 9]. However, control parameter tuning is still intractable. Hence, the control gains in PID scheme need to be calibrated carefully locally, which is mostly achieved by complicated experiments via trial-anderror approach during various operating conditions, and such process is both costly and time-consuming. Unfortunately, when harsh uncertain external condition causes the operating point of the engine to deviate farther from its calibrated condition, the well-tuned PID controller still cannot guarantee good property [4]. Recently, modern control theory and intelligent control method are applied to improve the adaptive and robust performance of the traditional PID control scheme in marine engine speed control. For instance, a self-turning PID controller based on BP neural network was designed for a largescale low-speed two-stroke marine diesel engine [16]; better control performance was achieved in simulation. Fuzzy control method was combined with PID to intelligently regulate the parameters of the speed controller online for a MAN B&M type diesel engine. The results showed the proposed fuzzy-PID controller possesses high anti-interference and strong robust abilities [17]. The 𝐻∞ control scheme was utilized as a tuning methodology for the PID speed regulator in [3] to guarantee the robustness against neglected dynamics in a propulsion marine engine and so on. However, we still underline that the application of MPC method in engine control domain is an unstoppable tendency, various up-to-date relevant articles are giving clue (e.g., [18– 21]), and just some issues as mentioned need to be solved furtherly. In many other industrial fields, some researchers started to pay attention to combine PID with MPC. In such applications, the tuning parameters of PID are obtained by MPC optimization law online to keep the advantages and alleviate the shortcomings in both methods. The control performance of these applications has been reported to be excellent. For example, PID was combined with predictive functional control (PFC) [22–25], synthesized with generalized predictive control (GPC) [26, 27], compounded with dynamic matric control (DMC) [28, 29], composited of some other predictive methods as shown in [30, 31].

Mathematical Problems in Engineering Up to now, there is no related study published in the field of marine diesel engine yet. Hence, it is deserved to study such kind of application in marine engine speed control, expecting to get good performance and reduce parameters adjustment. Apart from the above facts, for marine diesel engine speed control, temporally, it is reported that most of the control algorithms were verified via engine models. Some of these models were commonly used mean value engine models (MVEM) [2], and some were even simple low-order transfer function models [3, 32] or another kind of reduced complexity engine model [10]. In such situation, if a model could simulate the characteristics of the engine closer to the real one, it would help to make the verification of control algorithm more reliable and practical. Some innovative examples considered the discrete events in reciprocating engines, and hybrid models (the existence of discrete and continuous model) were studied to model the engine behaviors closer to the reality, such as [33, 34]. However, these models did not consider the whole engine system and only paid attention to intake manifold dynamic and the discrete events in torque generation, and they were merely limited at the idle mode. Some other important characteristics are ignored in such models, such as the working imbalance among different cylinders and the cycle delays, which have a significant effect in engine speed response and can influence the controller’s performance. Besides, cylinder-by-cylinder engine model (CCEM) also has been introduced into the engine control domain [35]. Although it can simulate engine more realistically, it requires in-cylinder pressure map, which has higher cost and is difficult to implement [36]. Motivated by the previous research and the challenges stated above, this paper develops a novel control scheme which combines the multiple model predictive functional control (MMPFC) with conventional PID for the marine diesel engine speed control, which markedly simplifies the parameter tuning work. The main contributions of this paper are listed as follows. First, we adopt the multiple model strategy, which aims at solving the high nonlinearity and various working conditions of the marine diesel engine. With a set of locally linearized predictive models to approximate the nonlinear system, multimodel approach is able to deal with the nonlinear system with wide working range [37, 38]. Second, the PFC theory is introduced to design the optimizer, which guarantees good tracking performance and robustness with fewer requirements in model structure information. Meanwhile, the calculation amount within the PFC is acceptable, so that the processing speed is fast [39]. The proposed controller is developed via combining the PFC with a classical incremental PID. Third, an online identification algorithm is introduced to obtain the open-loop gain, by which system robustness towards mismatched model is enhanced. Fourth, considering the drawbacks in the existing hybrid model and the difficulty to obtain the CCEM, as a compromise, an accurate cycledetailed hybrid nonlinear engine model is built in this study to evaluate the proposed controller. On the basis of such engine model, simulation results exhibit that the proposed controller with better performances compared with a welltuned conventional PID dealt with integral windup scheme.

Mathematical Problems in Engineering And the online identification algorithm is effective in improving the system robustness under disturbance loads. The rest of this paper is structured as follows. In Section 2, the combined MMPFC algorithm for engine speed control is described, and the modification method to update the model parameters online is discussed. In Section 3, a cycle-detailed hybrid nonlinear engine model is constructed for verifying the proposed controller, and the comparison between the proposed engine model and the classic MVEM is analyzed. In Section 4, the combined MMPFC method is illustrated in detail to apply in marine diesel engine speed control. And both steady and dynamic performances of the proposed controller are assessed by comparing with a well-tuned PID controller. In Section 5, a conclusion is given to sum up the whole work.

2. Controller Design 2.1. Description of the Predictive Multimodels for Marine Engine Speed Control. In order to design the control algorithm, the predictive multimodels must be obtained in a certain type. For diesel engine speed control, as shown in [40], the relationship between the control input (injection quality) and the system output (engine speed) is high nonlinear. It is difficult to get the accurate physical model, but fortunately there are simplified methods to obtain other types of model. When engine runs around a certain speed and load condition, as shown in [41–44], the complex engine model can be regarded as simple low-order linear model. The explanation and proof can be found in [44], where the identified result indicates that it is reasonable to adopt the first-order autoregressive (AR) model to represent the speed control modeling in a diesel engine when it works around a specific speed range. In this study, the FOPTD model is adopted as the predictive model in each locally linearized zone, which holds the equivalent structure as the mentioned first-order AR model. Because it captures the process gain (indicates the size and direction of the process variable response to a control move), overall time constant (describes the speed of the response), and effective dead time (states the delay before the response begins) of the process [37]. Furthermore, this kind of model can be identified easily from both simulation and experimental data, generally, from step response data [42]. The working condition of the engine would vary largely as the engine speed and load change. Although the estimation of the load is difficult, it is easy to measure the engine speed. Hence, the zone of local linearization is divided according to the engine speed, and the predictive multimodels of the engine for designing the proposed controller can be written as 𝐾𝑖 −𝜏𝑖 𝑠 𝑒 , 𝐺𝑖 (𝑠) = (1) 𝑇𝑖 𝑠 + 1 where 𝑖 = 1, 2, . . . , 𝑁 represents the different speed stage of the target engine under a certain load condition, and the speed range 𝑖 ∈ [𝑛𝑖𝑒 , 𝑛𝑒𝑖+1 ). 𝐺𝑖 (𝑠), 𝐾𝑖 , 𝑇𝑖 , 𝜏𝑖 are the

3 transfer function model between output and the control input, the process gain, the overall time constant, and the effective dead time during the engine speed range 𝑖, respectively. Remark 1. To upgrade model accuracy, the speed stages above should be divided more and cover all the speed ranges. For example, the whole speed range of the target engine is from 800 rpm to 2000 rpm in this study, it can be divided every 100 rpm or 50 rpm during the whole speed range, and the local model can be identified by step speed response in each stage. Note that the predictive multimodel in a certain speed stage is not unique, because different controller parameters will provide different result. In this study, the model is identified offline by using the data under a conventional PID controller. The parameters of such PID controller are obtained by simple 𝑍-𝑁 method. Hence, the engine speed controller can be designed on the basis of the identified models, which describe the basic dynamic for engine speed system in each speed range. 2.2. Multimodel Switching Method. In multimodel predictive control (MMPC), the switching scheduler is necessary to shift the local model (or local controller) for keeping stable transition among different operating conditions [45]. And the design of the switching scheduler directly affects the stability of the close-loop system [38]. In this study, because the predictive multimodels are gained based on the different engine speed stages, a switching method by referring to the present engine speed is proposed. As mentioned in (1), when the local model of the speed range 𝑖 ∈ [𝑛𝑖𝑒 , 𝑛𝑒𝑖+1 ) is gained, assuming that this local model is the model at the middle speed point (𝑛𝑒𝑖 + 𝑛𝑒𝑖+1 )/2, then 𝑁 local models belong to 𝑁 speed points are obtained. Linear interpolation is applied to calculate the present model parameters based on the 𝑁 local models by referring to the present engine speed. This can be realized easily by look-up map method. 2.3. Basic Combined MMPFC-PID Controller. For the predictive multimodels (1), ignoring the time delay, (1) can be discretized as follows: 𝑦𝑚 (𝑘) = 𝛼 (𝑘 − 1) 𝑦𝑚 (𝑘 − 1) + 𝐾 (𝑘 − 1) (1 − 𝛼 (𝑘 − 1)) 𝑢 (𝑘 − 1) ,

(2)

where 𝑦𝑚 (𝑘) is the output of the predictive model at sampling instant 𝑘, 𝛼(𝑘 − 1) = 𝑒−𝑇0 /𝑇(𝑘−1) . 𝐾(𝑘 − 1) and 𝑇(𝑘 − 1) are process gain and time constant calculated by look-up map method from the predictive multimodels at the sampling instant 𝑘 − 1. 𝑇0 is the sample and control time. The predictive output will be conducted based on the control input at sampling instant 𝑘, namely, 𝑢(𝑘). First, assume that the control input will keep the same within future 𝑝 (predictive horizon) steps, that is, 𝑢(𝑘 + 𝑝) = 𝑢(𝑘 + 𝑝 − 1) = ⋅ ⋅ ⋅ = 𝑢(𝑘).

4

Mathematical Problems in Engineering

Hence, the output predicted 1 step ahead of the predictive model from data at the sampling instant 𝑘 which is

where 𝑒(𝑘 + 𝑗) is the error between the corrected output and the output of the predictive model without time delay.

𝑦𝑚 (𝑘 + 1 | 𝑘) = 𝛼 (𝑘) 𝑦𝑚 (𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘)) 𝑢 (𝑘) . (3)

Remark 2. The cost function means getting the optimal solution within the predictive horizon 𝑝 rather than only at the predicted 𝑝 step as mentioned in [24]. By combining (5) with (8), 𝐽 can be written as

Then the output predicted 2 steps before the predictive model from data at the sampling instant 𝑘:

𝑝

𝑦𝑚 (𝑘 + 2 | 𝑘)

𝐽 = min ∑ (𝑦𝑟 (𝑘 + 𝑗)

= 𝛼 (𝑘) (𝛼 (𝑘) 𝑦𝑚 (𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘)) 𝑢 (𝑘))

(10)

𝑗=1

= 𝛼 (𝑘) 𝑦𝑚 (𝑘 + 1 | 𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘)) 𝑢 (𝑘 + 1)

𝑗

(4)

2

𝑗

− (𝛼 (𝑘) 𝑦𝑚 (𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘) ) 𝑢 (𝑘) + 𝑒 (𝑘 + 𝑗))) .

When 𝜕𝐽/𝜕𝑢(𝑘) = 0, the optimal control input 𝑢∗ (𝑘) is

+ 𝐾 (𝑘) (1 − 𝛼 (𝑘)) 𝑢 (𝑘)

𝑢∗ (𝑘)

= 𝛼 (𝑘)2 𝑦𝑚 (𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘)2 ) 𝑢 (𝑘) .

𝑝

Similarly, the output predicted 𝑝 steps ahead of the predictive model from data at the sampling instant 𝑘: 𝑦𝑚 (𝑘 + 𝑝 | 𝑘) = 𝛼 (𝑘)𝑝 𝑦𝑚 (𝑘) + 𝐾 (𝑘) (1 − 𝛼 (𝑘)𝑝 ) 𝑢 (𝑘) .

(5)

(6)

where 𝑦(𝑘) is the actual output of the plant at the sampling instant 𝑘, 𝑦correct (𝑘) is the corrected output of the predictive model at step k, 𝐷(𝑘) = 𝜏(𝑘)/𝑇0 , 𝜏(𝑘) is the time delay calculated by look-up map method from the predictive multimodels at the sampling instant 𝑘. The future reference trajectory 𝑦𝑟 (𝑘 + 𝑗) is set as 𝑦𝑟 (𝑘 + 𝑗) = 𝛽𝑗 𝑦 (𝑘) + (1 − 𝛽𝑗 ) 𝑐 (𝑘) ,

(7)

where 𝛽 represents the gentle factor of the reference; 𝑐(𝑘) is the set-point at the sampling instant 𝑘. Here, cost function 𝐽 is chosen as 𝑝

2

∑𝑗=1 (𝑦𝑟 (𝑘 + 𝑗) − (𝛼 (𝑘)𝑗 𝑦𝑚 (𝑘) + 𝑒 (𝑘 + 𝑗))) 𝑝 ∑𝑗=1

𝑗

𝐾 (𝑘) (1 − 𝛼 (𝑘) )

.

(11)

Conveniently, the incremental PID controller is proposed as follows: 𝑢 (𝑘) = 𝑢 (𝑘 − 1) + 𝐾𝑃 (𝑘) 𝑒1 (𝑘)

By taking account of the ignored time delay, the corrected output of the predictive model is given as follows: 𝑦correct (𝑘) = 𝑦 (𝑘) + 𝑦𝑚 (𝑘) − 𝑦𝑚 (𝑘 | 𝑘 − 𝐷 (𝑘)) ,

=

𝐽 = ∑ (𝑦𝑟 (𝑘 + 𝑗) − (𝑦𝑚 (𝑘 + 𝑗 | 𝑘) + 𝑒 (𝑘 + 𝑗))) ,

(8)

𝑒 (𝑘 + 𝑗) = 𝑦correct (𝑘) − 𝑦𝑚 (𝑘) ,

(9)

𝑗=1

+ 𝐾𝐼 (𝑘) (𝑒1 (𝑘) − 𝑒1 (𝑘 − 1)) + 𝐾𝐷 (𝑘) (𝑒1 (𝑘) − 2𝑒1 (𝑘 − 1) + 𝑒1 (𝑘 − 2)) ,

(12)

𝑒1 (𝑘) = 𝑐 (𝑘) − 𝑦 (𝑘) , where 𝑒1 (𝑘) presents the error between reference value and actual output of the plant at the sampling instant 𝑘, 𝐾𝑃 (𝑘), 𝐾𝐼 (𝑘), 𝐾𝐷(𝑘) are PID parameters at the sampling instant 𝑘. The PID controller can be rewritten as 𝑢 (𝑘) = 𝑢 (𝑘 − 1) + w (𝑘)𝑇 E (𝑘) w (𝑘) , 𝑇

w (𝑘) = [𝑤1 (𝑘) , 𝑤2 (𝑘) , 𝑤3 (𝑘)] ,

(13) (14)

𝑇

E (𝑘) = [𝑒1 (𝑘) , 𝑒1 (𝑘 − 1) , 𝑒1 (𝑘 − 2)] ,

(15)

𝑤1 (𝑘) = 𝐾𝑃 (𝑘) + 𝐾𝐼 (𝑘) + 𝐾𝐷 (𝑘) , 𝑤2 (𝑘) = −𝐾𝑃 (𝑘) − 2𝐾𝐼 (𝑘) ,

(16)

𝑤3 (𝑘) = 𝐾𝐷 (𝑘) . By solving (11) and (13) we can get w(𝑘) as

𝑝

w (𝑘) =

(∑𝑗=1 ((𝑦𝑟 (𝑘 + 𝑗) − 𝐾 (𝑘) (1 − 𝛼 (𝑘)𝑗 ) 𝑢 (𝑘 − 1) + 𝛼 (𝑘)𝑗 𝑦𝑚 (𝑘) + 𝑒 (𝑘 + 𝑗)))) 𝑝

(∑𝑗=1 𝐾 (𝑘) (1 − 𝛼 (𝑘)𝑗 ) E𝑇 (𝑘) E (𝑘) + 𝜆)

Here, a variable 𝜆 is introduced, and thus the denominator will never become zero. The optimized PID parameters 𝐾𝑃 (𝑘), 𝐾𝐼 (𝑘), 𝐾𝐷(𝑘) can be calculated through (16), and they are

.

(17)

𝐾𝐷 (𝑘) = 𝑤3 (𝑘) , 𝐾𝑃 (𝑘) = −𝑤2 (𝑘) − 2𝐾𝐷 (𝑘) , 𝐾𝐼 (𝑘) = 𝑤1 (𝑘) − 𝐾𝑃 (𝑘) − 𝐾𝐷 (𝑘) .

(18)


5

e1(k)

c(k) +

ym (k + 1 | k) · · · ym (k + p | k) e(k + j)

KD(k)

KI(k)

KP(k)

FPC min J

FPC-PID controller

Predictive Multi-models

u(k)

Diesel engine

y(k)

−

Figure 1: Basic control structure diagram of the MMPFC-PID method.

Here, the PID parameters are optimized by multimodel PFC method. This method can be called MMPFC-PID. And, about the practicability, robustness and stability of the proposed controller can be found similarly in [26]. The basic control structure diagram is illustrated in Figure 1. Provided with proper relative accurate predictive multimodels and basic parameters in the MMPFC-PID method, the proposed controller can optimize the PID parameters in each control step. 2.4. Adaptive Open-Loop Gain Associated MMPFC-PID Controller. Although it has been proved by many previous researchers that the combined PFC-PID controller can keep good performance when the predictive model changes in a certain scale [22–25], it is a fact that the mismatched model would make the control performance deteriorate if the predictive model deviates far from the designed condition [37, 39, 46]. For the marine main engine, the load condition varies more extensively. Particularly, when the wave causes the emergence of the propeller disk from the water, the load of the engine would change violently in such process [47]. Most intuitively, it causes the open-loop gain of the engine speed control modeling to change drastically. The open-loop gain would be far away from its value in the identified condition. In this situation, the basic MMPFCPID cannot keep good dynamic performance. Hence, the online identification algorithm is necessary for alleviating the influence of model deviation. In this subsection, the feasible way to estimate the open-loop gain online and its application in MMPFC-PID controller to modify the parameters in FOPTD model will be discussed. Assume that the diesel engine can be regarded as linear system in each linearized zone. For the linear system, it can be shown by difference equation as follows: 𝑦 (𝑘) = 𝛼1 𝑦 (𝑘 − 1) + ⋅ ⋅ ⋅ + 𝛼𝑛 𝑦 (𝑘 − 𝑛) + 𝛽0 𝑢 (𝑘 − 𝑘0 ) + ⋅ ⋅ ⋅ + 𝛽𝑛−1 𝑢 (𝑘 − 𝑘0 − 𝑛 + 1) + 𝑒 (𝑘) .

(19)

Select the parameter estimation equation of the object as 𝑦 (𝑘) = 𝜙𝑇 (𝑘) 𝜃 (𝑘) + 𝑒 (𝑘) , 𝜙𝑇 (𝑘) = [𝑦 (𝑘 − 1) , . . . , 𝑦 (𝑘 − 𝑛) , 𝑢 (𝑘 − 𝑘0 ) , . . . ,

𝑢 (𝑘 − 𝑘0 − 𝑛 + 1) , ̃𝑒 (𝑘 − 1) , . . . , ̃𝑒 (𝑘 − 𝑛)] , 𝜃𝑇 (𝑘) = [𝛼1 (𝑘) , . . . , 𝛼𝑛 (𝑘) , 𝛽0 (𝑘) , . . . , 𝛽𝑛−1 (𝑘) , 𝑐1 (𝑘) , . . . , 𝑐𝑛 (𝑘)] ,

(20) where the estimated residual can be written as ̃𝑒 (𝑘 − 1) = 𝑦 (𝑘 − 1) − 𝜙𝑇 (𝑘 − 1) 𝜃 (𝑘 − 2) .

(21)

The parameters estimation algorithm is given as 𝜃 (𝑘) = 𝜃 (𝑘 − 1) +

(22) 𝜆 1 𝜙 (𝑘) [𝑦 (𝑘) − 𝜙𝑇 (𝑘) 𝜃 (𝑘 − 1)] , 𝜙 (𝑘) 𝜙 (𝑘) +𝜆 2 𝑇

where 𝜆 1 is the forgotten factor; 𝜆 2 is a constant to avoid the denominator equal to zero. Then the open-loop gain of the object is estimated by 𝐾∞ =

𝑦 (∞) ∑ 𝛽𝑖 , = 𝑢 (∞) 1 − ∑ 𝛼𝑖

(23)

where 𝑦(∞) and 𝑢(∞) mean the system output and control input under steady state, respectively. Remark 3. This method does not need matrix computation. Hence, it is easy to guarantee convergence and it can be realized easily in engineering perspective. Even though the exact model of the marine main engine is unknown or uncertain, it is possible to estimate the open-loop gain once a relative high-order estimation model is chosen. Due to the existence of noise in system output and control input, Kalman filter is adopted to dispose the original data to get smooth estimation result. The process gain in FOPTD model will be replaced by the identified open-loop gain in each control step. Note that although the time constant in FOPTD model for engine speed control modeling cannot be identified easily as the process gain, it has a positive correlation with the process gain in the most conditions. This will be illustrated later in this paper. To simplify the algorithm, the time constant in FOPTD model

6

Mathematical Problems in Engineering ym (k + 1 | k) · · · ym (k + p | k) e(k + j) Predictive Multi-models K∞

KD(k)

KI(k)

KP(k)

FPC min J

Open-loop Gain Estimation e1(k)

c(k) +

FPC-PID controller

u(k)

Diesel engine

y(k)

−

Figure 2: Modified control structure diagram of the MMPFC-PID controller.

can be modified by using the change ratio of the process gain shown as follows: 𝐾 (𝑘) 𝑇 (𝑘) , 𝑇󸀠 (𝑘) = ∞ 𝐾 (𝑘)

Table 1: Target marine diesel engine specifications. Engine type

(24)

where 𝑇󸀠 (𝑘) means the modified time constant at the sampling instant 𝑘, 𝐾∞ (𝑘) is the estimated open-loop gain at the sampling point 𝑘, 𝑇(𝑘) and 𝐾(𝑘) represent the time constant and process gain calculated by look-up map from the predictive multimodels at the sampling instant 𝑘. The basic MMPFC-PID method can be modified with the identification of the open-loop gain online, as shown in Figure 2. It can be called AGMMPFC-PID (adaptive openloop gain MMPFC-PID) method. When the engine load changes largely, the predictive multimodels can be modified to adapt to the load changing. As a result, the adaptive ability in MMPFC-PID controller can be improved extensively.

3. Engine Model As mentioned above, a more accurate diesel engine model is necessary for testing the speed control algorithm. In this part, a cycle-detailed hybrid nonlinear engine model which incorporates both discrete and continuous phenomena will be given for the target engine. In such model, the engine is considered as an integrated system. The torque generation is modeled by pulse indicated torque which derives from the MVEM method. Furthermore, the cylinder-bycylinder variation and the cycle delays are under consideration. The specifications of the target engine are listed in Table 1. The thermodynamic volumes in the marine diesel engine model are shown in Figure 3. According to the schematic, the engine model consists of five subsystems, namely, intake manifold, exhaust manifold, cylinders, intercooler, and turbocharger. We will explain the model in two main parts, the discretetime and continuous-time processes.

Displacement Number of cylinders Bore/stroke Rated power Compression ratio Intake manifold

Water cooling, inline, 4-stroke, direct injection 12.7 L 6 180/200 385 Kw (2000 rpm) 17.0 : 1 Turbo charged intercooled

3.1. Discrete-Time Process. To determine the timing sequence of the four strokes in each cylinder during its working cycle, the crank-angle (CA) signal 𝜑 is introduced by the integration of the engine speed; that is, 𝜑 = mod (∫ 6𝑛𝑒 𝑑𝑡, 720) ,

(25)

where 𝑛𝑒 is engine speed; operator “mod” represents modulus. Note that the related position for all the cylinders is steady, for the target engine, 6-cylinder 4-stroke inline engine, the cylinders are distributed on the crankshaft every 120∘ CA. For each cylinder, the torque generation mechanism and the in-cylinder evolution process can be illustrated via finitestate machines (FSMs) [33, 34]. As shown in Figure 4, the torque is only generated in the process from “𝐶” to “𝐸.” To make use of the advantage of the MVEM method, assuming the torque is kept constant during this process, we substitute the mean torque in the process from “𝐶” to “𝐸” for the mean value in the whole working cycle. Figure 4 also reveals the fact that the intake process and exhaust process happen in a certain timing sequences, which will be used to illustrate the cycle delays later.


7 360 360 {1 (𝑖 − 1) < 𝜑 < (𝑖 − 1) + 𝜑𝑓 𝑁𝑐 𝑁𝑐 𝑃durtion = { {0 otherwise

Intercooler

Intake Manifold W31

T1 , P1 1

T3 3

+ ECU

Wf

W1e

𝑖 = 𝑓 (𝑛𝑒 , 𝜆𝑖 ) , 𝜂𝑖𝑔

Wc3

−

pc , Tc

c

Cylinders

Turbocharger

COMP

ne , Mig

TURB e

t We2

W2t

where 𝑃durtion is block pulse function; it makes the gross torque zero except during the expansion phase; as can be explained in (27), 𝑊𝑓𝑖 is the fuel mass flow rate injected into the cylinder 𝑖, 𝑞𝐻𝑉 represents the low calorific value of fuel, 𝑖 is the gross indicated efficiency of the cylinder 𝑖, it is the 𝜂𝑖𝑔 function of the engine speed and air to fuel ratio (AFR), and, for the cylinder 𝑖, it is 𝜆𝑖 , 𝜑𝑓 is the lasting angle of powering, and for the target engine it is 180∘ CA, and 𝑁𝑐 means the number of total cylinders. The relationship between 𝑊𝑓𝑖 and the control input 𝑚𝑓 (injection quality per stroke per cylinder) is 𝑛𝑒 ⋅ 𝑁𝑐 ⋅ 𝑚𝑓

(29) ⋅ 10−6 . 120 The total gross indicated torque of all the cylinders 𝑀𝑖𝑔 is

T2 , P2

𝑁𝑐

Exhaust Manifold

Figure 3: Schematic diagram of the marine diesel engine in a ship for propulsion using. Note. 𝑊𝑖𝑗 denotes the mass flow rate from volume “𝑖” to volume “𝑗.” 𝑊𝑓 represents the whole mass flow rate injected into the cylinders. 𝑇1 and 𝑃1 are temperature and pressure in intake manifold, respectively. 𝑇2 and 𝑃2 are temperature and pressure in exhaust manifold, respectively. 𝑇3 is the coolant temperature.𝑇𝑐 and 𝑃𝑐 are temperature and pressure at the outlet of the compressor, respectively. Exhaust condition calculation

𝑖 . 𝑀𝑖𝑔 = ∑ 𝑀𝑖𝑔

I

(30)

1

3.2. Continuous-Time Process 3.2.1. Intake Manifold. According to the first law of thermodynamics and the properties of ideal gas, the pressure state equations for intake manifold can be gained as follows: 𝑑𝑝1 𝛾𝑅 (𝑊𝑐1 𝑇𝑐1 − 𝑊1𝑒 𝑇1 ) , = 𝑑𝑡 𝑉1 𝑊1𝑒 =

(31)

𝑁𝑐

∑ 𝑊1𝑒𝑖 . 1

The mass flow rate of the intake charger for the individual cylinder 𝑖, 𝑊1𝑒𝑖 , is

Intake condition calculation

E

(28)

𝑊𝑓𝑖 =

2

H

(27)

𝑊1𝑒𝑖 = 𝜂V ⋅

𝑝1 𝑛𝑒 𝑉 ⋅ 𝑑, 120𝑅𝑇1 𝑁𝑐

(32)

where 𝜂V is the volumetric efficiency of the engine; 𝑉𝑑 is the displacement of the engine. Note that, in this equation, the value of 𝑝1 , 𝑛𝑒 , 𝑇1 must be at the moment when 𝑊1𝑒𝑖 is calculated, or they cannot donate the mass flow rate of the cylinder 𝑖.

C

Torque generation

Figure 4: In-cylinder evolution process.

Furthermore, to model the cylinder-by-cylinder variation in cylinders, a factor 𝜉𝑖 is introduced, which is defined as the cylinder-by-cylinder variation of the cylinder 𝑖, requiring 0 ≤ 𝜉𝑖 ≤ 1, when the gross indicated torque is calculated. 𝑖 Hence, for the cylinder 𝑖, the gross indicated torque 𝑀𝑖𝑔 can be written as 𝑖 𝑀𝑖𝑔

= 𝑃durtion

30 ⋅ 𝑊𝑓𝑖 ⋅ 𝑞

𝐻𝑉

𝜋 ⋅ 𝑛𝑒

⋅𝜂𝑖𝑖𝑔 720 𝜉, 𝑁𝑐 ⋅ 𝜑𝐹 𝑖

(26)

3.2.2. Exhaust Manifold. Similar to the intake manifold, the pressure state equations in exhaust manifold can be described as 𝑑𝑝2 𝛾𝑅 (𝑊𝑒2 𝑇𝑒2 − 𝑊2𝑡 𝑇2 ) , = 𝑑𝑡 𝑉2 𝑁𝑐

𝑊𝑒2 = ∑ 𝑊𝑒2𝑖 , 1

𝑊𝑒2𝑖

= 𝑊1𝑒𝑖 + 𝑊𝑓𝑖 .

(33)

8

Mathematical Problems in Engineering For simplicity, the following assumption is made: 𝑊1𝑒 = 𝑊31 = 𝑊𝑐3 .

(34)

3.2.3. Intercooler. Ignoring the dynamic in intercooler, the steady state model can be modeled as 𝑇1 = 𝑇𝑐 (1 − 𝜂3 ) + 𝑇3 𝜂3 ,

(35)

where 𝜂3 means the intercooler efficiency, which can be obtained by experimental data. The pressure loss in the intercooler is given as Δ𝑃 = 𝑃𝑐 − 𝑃1 = 𝑘3

𝑊𝑐3 2 𝑅𝑇 , 𝑃𝑐

(36)

where 𝑘3 is the friction factor in the intercooler while the gas is passing by. 3.2.4. Turbocharger. Turbocharger incorporates two components, the turbo and the compressor. Due to the high nonlinearity in turbocharger, it is difficult to derive an accurate physical model. In general, the map based model is adopted by using of the steady state compressor mass flow map and efficiency map, which are related to the turbocharger speed and pressure ratio. The rotational dynamics of the turbocharger can be written as 𝑃 𝜂 − 𝑃𝑐 𝑑𝑛𝑡 ), = 1000 ( 𝑡 𝑚 (37) 𝑑𝑡 𝐽𝑡 𝑛𝑡 where 𝑛𝑡 is the turbine speed, 𝑃𝑡 is the turbine power produced by exhaust, 𝑃𝑐 is the power which the compressor consumes, 𝐽𝑡 is the inertia, and 𝜂𝑚 is mechanical efficiency from turbine to compressor. 𝑃𝑡 can be given as 𝑃𝑡 = 𝑊2𝑡 𝑐𝑝𝑒 𝑇𝑡∗

1 (𝜋 (𝛾𝑒 −1)/𝛾𝑒 − 1) , 𝜂𝑡 𝑡

(38)

where 𝑐𝑝𝑒 is the specific heat capacity for exhaust gas at constant pressure, 𝑇𝑡∗ is the temperature of the inlet in turbine under stagnation condition, 𝜂𝑡 is the turbo efficiency, 𝜋𝑡 is the pressure ratio across the turbo, and 𝛾𝑒 is isentropic exponent of exhaust gas. 𝑃𝑐 can be described as 𝑃𝑐 = 𝑊𝑐3 𝑐𝑝𝑎 𝑇𝑐∗

1 (𝜋 (𝛾𝑎 −1)/𝛾𝑎 − 1) , 𝜂𝑐 𝑐

3.4. Load Torque. According to [2, 3] and the references therein, the load torque for a propulsion marine main engine can be normalized and described as 𝑀load = 𝑘load 𝑛𝑒 2 ,

(41)

where parameter 𝑘load represents the integrative effect from the properties of the ship (size, shape, etc.) and the load condition. The various types of bounded disturbance (disturbances in fuel injection, combustion, load, etc.) are all taken into consideration as 𝑀noise , which is gotten by a bounded noise signal. The load coming from wave is another crucial load disturbance in marine main engine. It can be defined as 𝑀wave and simulated via low-frequency sine signal [2, 48]. 3.5. Engine Rotational Dynamic Model. Considering the whole effect from the all cylinders, the engine rotational dynamic model can be written as 𝐽𝑒

𝑑𝑛𝑒 = 𝑀𝑖𝑔 𝑑𝑡

(42)

− (𝑀𝑝 + 𝑀𝑓 + 𝑀load +𝑀noise + 𝑀wave ) , where 𝐽𝑒 is the equivalent rotary inertia for the diesel engine with consideration of its connections (propellers, etc.), 𝑀𝑓 is the friction torque, and 𝑀𝑝 is the pumping torque. 𝑀𝑝 and 𝑀𝑓 can be calculated by the following equations:

(39)

where 𝑐𝑝𝑎 is the specific heat capacity for air at constant pressure, 𝑇𝑐∗ is the temperature of the inlet in compressor under stagnation condition, 𝜂𝑐 is the compressor efficiency, 𝜋𝑐 is the pressure ratio across the compressor, and 𝛾𝑒 is isentropic exponent of air. The turbo-lag has a large effect during the dynamic process, it can be modeled as a first-order lag. The dynamic of the power demanded by the compressor can be written as 𝑑𝑃𝑐 1 = − (𝑃𝑡 𝜂𝑚 − 𝑃𝑐 ) , 𝑑𝑡 𝜏𝑐

3.3. Engine Cycle Delays. As shown in Figure 4, the evolution in cylinder happens in different time, which leads to cycle delays of the engine model. These delays can be descripted in either CA domain or the time domain; here we define it in CA domain. We choose the moment at the end of the intake stroke as the reference point. The CA duration of a single stroke is 180∘ CA for a 4-stroke engine. Through Figure 4, we know that the fuel injection is delayed by 180∘ CA, and 𝑖 and 𝑊𝑓𝑖 , the power stroke relevant variables, such as 𝑀𝑖𝑔 ∘ are delayed by 2 × 180 CA, and the exhaust stroke relevant variables are delayed by 3 × 180∘ CA. These delays can be realized by setting the updating moments for these related parameters in each cylinder based on the CA domain.

𝑀𝑝 =

𝑉𝑑 (𝑃 − 𝑃1 ) , 4𝜋 2

𝑀𝑓 =

𝑉𝑑 5 10 (𝐶𝑓1 𝑛2 𝑒 + 𝐶𝑓2 𝑛𝑒 + 𝐶𝑓3 ) , 4𝜋

(43)

where 𝐶𝑓1 , 𝐶𝑓2 , 𝐶𝑓3 are coefficients to decide friction torque, which can be obtained by experiment. For all the symbols mentioned above, some of them are not annotated; the particular explanation can be found in Figure 3 and the annotation therein.

(40)

where 𝜏𝑐 is the time constant; for simplicity, we assume it to be a constant.

3.6. Comparison between the Cycle-Detailed Hybrid Nonlinear Engine Model and the MVEM. The comparisons of engine speed responses and indicated torque between the proposed


9 3000 2500 Indicated torque (Nm)

Speed (rpm)

1600

Zoom

1500

1610

1400 1600 1597

2

4

6

4 Time (s)

6

2000 1500 1000 500

1300 0

2

0

8

0

2

MVEM Proposed engine model

4 Time (s)

6

8

MVEM Proposed engine model

(a)

(b)

Figure 5: The comparisons of engine models in speed responses (a) and indicated torque (b). Table 2: The notation for different speed references. Process

800 rpm to 1200 rpm

1200 rpm to 1600 rpm




1200 rpm to 800 rpm

Notation

I

II

III

IV

V

VI

model and the MVEM are shown in Figure 5. Note that the speed responses in Figure 5 are obtained under the same conditions for both models, which means that the noise load, the load factor, and the parameters of controller are consistent in both cases. As shown in Figure 5(a), more detailed system performance can be observed when the proposed model is adopted. Both the steady and the transient performances in the proposed hybrid nonlinear engine model seem to be inferior to them in the MVEM, because the uncertainties and nonlinearities have been considered more sufficiently in the proposed model. From Figure 5(b), we know that the pulse indicated torque in engine can be shown through the proposed model. Although it is calculated by modified MVEM method as mentioned, it can model the discrete indicated torque without more complex model structure. As we can see, the mean effect of the pulse indicated torque is the same as that in the MVEM (the red dashed line is in the middle position of the green dashed line). Remark 4. Note that the MVEM was verified in author’s previous study [2]. In Figure 5(a), we know that the speed responses between the two models are the same broadly; this also can be regarded as the verification of the proposed model. Limited by the space, here we do not discuss more about it; for more details refer to [2]. To sum up the diesel engine model from the perspective of control, the engine rotational dynamic model (42) can be rewritten in the form of state space:

Table 3: The notation for different load conditions. Load condition

𝑘load = 0.00025

𝑘load = 0.00015

𝑘load = 0.00035

Notation

Case 1

Case 2

Case 3

𝑥̇ = 𝑓 (𝑥, 𝑢) 𝑥 + 𝑔 (𝑥, 𝑢) 𝑢 + 𝜔, 𝑦 = 𝑥,

(44)

where 𝑥 = 𝑛𝑒 is the state variable, control input 𝑢 = 𝑚𝑓 , system output 𝑦 means the engine speed, and 𝜔 represents the additive disturbance, primarily including model uncertainties and unmodeled dynamics.

4. Simulation Experiments and Results In this section, the simulation experiments will be designed based on the proposed engine model. To compare the system performance for the MMPFCPID controller and the PID controller in a wide engine working range, the reference speed is decided to include a series of step functions to cover the engine speed range from 800 rpm to 2000 rpm. Specifically, as shown in Table 2, initiate from the idle speed at 800 rpm to the rated speed 2000 rpm; both acceleration (marked as I, II, and III) and deceleration (marked as IV, V, and VI) processes are set. The three load factors are considered as shown in Table 3; the load factor 𝑘load is designed to be 0.00025 in Case 1, and

10

Mathematical Problems in Engineering 60

1.5

50

1.4

Overshoot (rpm)

Setting time (s)

40 1.3

1.2

30

20 1.1

1

10

1

2

3

4

0

1

2

3

ki = 0.2 ki = 0.3 ki = 0.4 ki = 0.5 ki = 0.6

4

kp

kp ki = 0.7 ki = 0.8 ki = 0.9 ki = 1.0 (a)

ki = 0.2 ki = 0.3 ki = 0.4 ki = 0.5 ki = 0.6

ki = 0.7 ki = 0.8 ki = 0.9 ki = 1.0 (b)

Figure 6: The demonstration of tuning PID parameters and the effects of 𝑘𝑝 and 𝑘𝑖 on setting time (a) and on overshoot (b).

when it has a deviation of −40% and +40%, Case 2 (𝑘load = 0.00015) and Case 3 (𝑘load = 0.00035) are obtained. The considered load cases cover a wide range of external load. Both controllers are tested with the mentioned different reference speed signals in each load case to cover the engine operating range extensively. The control performance comparisons of both controllers in these designed cases are carried out by comparing the indices in setting time (𝑡𝑠 ), overshoot percentage (𝑀𝑝 ), steady state speed fluctuation (|Δ𝑛𝑒 |), integral absolute error (IAE), and time-multiplied absolute error (ITAE). Furthermore, the capacity of resisting disturbance for the proposed controller should be evaluated. We choose the 2000 rpm case to execute such test, where the engine is the rated condition and the output power is the maximum. And the disturbance load incorporates two main types, that is, wave load and the mutation load. 4.1. Parameters Design of Controllers 4.1.1. Parameters Design of PID Controller. Because of the nonlinearity and the various working ranges in marine diesel engine, one set of PID parameters cannot keep good performances at different speed stages, different load conditions, and different processes (acceleration and deceleration). Hence, the PID parameters are tuned under Case 1 (regarded as the standard condition) at normal used highspeed condition (between 1600 rpm and 2000 rpm). In such

condition, the PID parameters are optimized via trial-anderror approach and deal with integral windup scheme. Figure 6 demonstrates the effects of basic PID parameters on the performance in setting time and overshoot. There is a trade-off relationship between setting time and overshoot while the proportionality coefficient (𝑘𝑝) is in a reasonable range (from 2 to 4, as shown in Figure 6(a)). When the proportionality coefficient and integral coefficient (𝑘𝑖) vary in the range from 3 to 4 and from 0.5 to 1.0, respectively, a proper performance can be obtained. At the same time, it also needs to make a compromise in the performances among acceleration, deceleration, and steady state processes. Furthermore, it is necessary to take the effect of differential coefficient (𝑘𝑑) into account. Finally, the PID parameters are decided as 𝑘𝑝 = 3.6, 𝑘𝑖 = 0.7, and 𝑘𝑑 = 0.4. From the optimization process above, we know that it is tiring and difficult. 4.1.2. Parameters Design of MMPFC-PID Controller. As for the proposed MMPFC-PID controller, the parameters also are designed under Case 1. To gain the basic input/output (injection quality/engine speed) data for the identification purpose, a closed-loop controller is required [44], because diesel engine is open-loop marginally stable [7, 49]. The closed-loop controller is chosen to be the PID with simple parameters by 𝑍-𝑁 approach. With the help of such controller, the input/output data is obtained by setting the engine


11

Table 4: The parameters of the identified predictive multimodels. Model parameters

850

950

1050

1150

𝑇𝑖 𝐾𝑖 𝜏𝑖

2.78 16.39 0.081

2.65 16.12 0.075

2.67 15.79 0.071

2.47 15.42 0.068

1250

Speed (rpm) 1350 1450

1550

1650

1750

1850

1950

2.40 14.95 0.063

2.26 14.41 0.058

2.23 13.31 0.045

2.00 11.37 0.041

2.19 12.05 0.033

2.19 11.51 0.024

1.44 10.93 0.015

2.10 13.92 0.055

2200 2002 2000

2000 1800 Speed (rpm)

1400

1600

Zoom 4

1520 18.5 19 19.5

13.5 14 14.5

Zoom 2

1600

Zoom 3

1602 1600

1200

Zoom 5

1120 23.5 24 24.5

8.5 9 9.5

1200 1000

1206 1200

800 600

Zoom 1 800

720 29

3.5 4 4.5

3

Zoom 6

30

8

13

18 Time (s)

23

28

33

8

13

18 Time (s)

23

28

33

Injection quality (mg/str.)

Reference PID MMPFC-PID 300 200 100 0

3

PID MMPFC-PID

Figure 7: Step speed responses and the control input in Case 1.

speed with step response every 100 rpm between 800 rpm and 2000 rpm. According to the front contents, the identified predictive multimodels can be shown in Table 4. In general, there is a positive correction between the process gain and the time constant. It provides a way to understand the modification method of the time constant in previous section. Other parameters for the MMPFC-PID controller are designed as follows: 𝛽 = 0, 𝜆 = 1. The predictive horizon 𝑝 affects the dynamic performance of the MMPFC-PID controller, which implies that a balance between the stability and dynamic response should be considered when we design 𝑝. We decide it to be changed with the variation of the time delay, 𝑝 = 𝐷(𝑘) + 10. Note that the smooth factor 𝛽 is designed to be 0 for getting an equitable comparison to the conventional PID controller, as when 𝛽 = 0 it means the set-point does not deal with smoothing method and can get

the fastest response. And both controllers are with the same sample and control time, 𝑇0 = 0.01 s. 4.1.3. Parameters Design of AGMMPFC-PID Controller. As for the AGMMPFC-PID controller, the basic parameters are the same as those in the MMPFC-PID. And the parameters in open-loop gain estimation process are designed as follows: the order of the engine system is chosen to be 3, forgotten factor is set as 𝜆 1 = 0.95, and constant 𝜆 2 = 0.5. 4.2. Simulation and Analysis 4.2.1. Speed Tracking Performance under Step Response Conditions. The responses of the engine speed and the corresponding control input for the mentioned Case 1, Case 2, and Case 3 are shown in Figures 7, 8, and 9, respectively. The according performance comparisons for both controllers

12

Mathematical Problems in Engineering 2200 1600

2000 13.5 14 14.5 Zoom 2

1800 Speed (rpm)

Zoom 3

2010

2000

Zoom 4

1500 18.5 19 19.5

1600

1620

1200

1400

1600 8.5 9 9.5

1100

Zoom 5

24

25.5

1200 1000

1230

800

Zoom 1 800

1200

600

3.5 4 4.5 8

3

Zoom 6

700 29.5

30.5

13

18 Time (s)

23

28

33

13

18 Time (s)

23

28

33



3

8

PID MMPFC-PID

Figure 8: Step speed responses and the control input in Case 2. 2200 2004 2000

2000

Speed (rpm)

1800 1600 1400

1602 1600

Zoom 3

Zoom 4 1600 1590 18.3 19

14.5 15.2

Zoom 2

Zoom 5 1200 1120 23.5 24 24.5

8.5 9 9.5

1200 1000

1204

800 600

Zoom 1

Zoom 6 800

1200

720 29 29.5 30

3.5 4 4.5

3

8

13

18 Time (s)

23

28

33

8

13

18 Time (s)

23

28

33



3

PID MMPFC-PID

Figure 9: Step speed responses and the control input in Case 3.


13 3

2.5

ts (s)

2

1.5

1

0.5 I

II

III

IV

V

VI

Process Case 1 MMPFC-PID Case 1 PID Case 2 MMPFC-PID

Case 2 PID Case 3 MMPFC-PID Case 3 PID

Figure 10: Comparisons of speed tracking performance in the index of setting time (𝑡𝑠 ) for all the tested processes and load factors. 12

10

Mp (%)

8

6

4

2

0 I

II

III

IV

V

VI



Figure 11: Comparisons of speed tracking performance in the index of overshoot percentage (𝑀𝑝 ) for all the tested processes and load factors.

in the criteria of setting time (𝑡𝑠 ), overshoot percentage (𝑀𝑝 ), steady state speed fluctuation (|Δ𝑛𝑒 |), integral absolute error (IAE), and time-multiplied absolute error (ITAE) are illustrated in Figures 10, 11, 12, 13, and 14, respectively. In Case 1 (shown in Figure 7), the most apparent trait is that the conventional PID controller cannot provide good

performance in reducing overshoot percentage during the deceleration processes (Figure 7: Zoom 4, Zoom 5, and Zoom 6). Detailed information can be found in Figure 11. Although the PID controller is well-tuned in the process IV, the overshoot percentage is still more than 4% (shown in Figure 10, the red line marked with stars). Besides, the

14


  Δne  (rpm)

5

4

3

2

1 800

1200 1600 Steady state speed (rpm) Case 1 MMPFC-PID Case 1 PID Case 2 MMPFC-PID

2000


Figure 12: Comparisons of speed tracking performance in the index of steady state speed fluctuation (|Δ𝑛𝑒 |) for all the tested processes and load factors. 1100

900

IAE

700

500

300

100

I

II

III

IV

V

VI



Figure 13: Comparisons of speed tracking performance in the index of integral absolute error (IAE) for all the tested processes and load factors.

overshoot percentages are about 7% and 10% in the processes V and VI (shown in Figure 11, the red line marked with stars). On the contrary, in the MMPFC-PID controller, the overshoot percentage values are kept less than 1% or even zero

in all the acceleration and deceleration processes (shown in Figure 11, the red line marked with squares). The preponderance in restricting overshoot for the MMPFC-PID controller as mentioned in Case 1 is still


15

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Figure 14: Comparisons of speed tracking performance in the index of time-multiplied absolute error (ITAE) for all the tested processes and load factors.

inherited when it comes to Case 2 and Case 3. Although the load factors vary ±40%, the overshoot percentages in all the processes are kept less than 2% (shown in Figure 11, green and blue lines marked with squares). But for PID, in Case 2 (shown in Figure 8), the overshoot percentages get worse, which are around 6%, 8%, and 11% in the processes IV, V, and VI (shown in Figure 11, green line marked with stars), respectively; in Case 3 (shown in Figure 9), the overshoot percentages are still inferior to them in the MMPFC-PID; they are almost 7% and 10.5% in processes V and VI (shown in Figure 11, blue line marked with stars), separately. Furthermore, as the indices in steady state speed fluctuation and time-multiplied absolute error can indicate the steady state traits and the indices in setting time and integral absolute error represent the dynamic characteristics [50], it is can be observed from Figures 10, 12, 13, and 14 that the same color lines marked with squares are clearly below the lines marked with stars. Hence, it is evident that the MMPFC-PID controller has better steady and dynamic performances in all the tested processes. According to the analysis above, the proposed MMPFCPID controller has better adaptive ability than the conventional well-tuned PID. It also can be deduced from the control input (injection quality) curves, shown in Figures 7–9, that the control input in the proposed controller is adjusted more quickly. It is because the PID parameters in the proposed controller are optimized by multimodel PFC method in each control step rather than keeping the same values in the welltuned PID controller. It should be noted that the parameters variations of the proposed controller are not given in this study, because they

Table 5: Performance Comparisons under wave load. Index IAE |Δ𝑛𝑒 | (rpm)

AGMMPFC-PID 70.07 5.4

Controller MMPFC-PID 106.52 6.2

PID 113.90 6.8

are totally dynamic optimization processes without a strong regularity. They were also never illustrated in the similar applications in other different fields [22–25, 27]. 4.2.2. Speed Governing Performance under Disturbance Load Conditions. The overall speed responses of the marine diesel engine under the wave load and mutation load conditions are exhibited in Figure 15. The system performance in different disturbance loads are enlarged in Figure 16. The detailed load changing is shown in Figure 17(a). During the time from 0 s to 10 s, the engine is working under 100% propulsion load condition and the same time disturbed by wave load. From 10 s to 15 s, both propulsion load and wave load are removed totally, but the noise load is kept. And in the last 5 s the total propulsion load and wave load are added in again. As shown in Figure 16(a), under wave load disturbance situation, the response of the AGMMPFC-PID is better than the performance given by the MMPFC-PID and PID. More details are shown in Table 5. The AGMMPFC-PID method has the smallest values in steady state speed fluctuation

16

Mathematical Problems in Engineering 2100 Sudden decrease 100% load

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Figure 15: The overall engine speed response in disturbance loads.

Table 6: Performance comparisons under mutation loads. Index Speed variation (abs. rpm) Recovery time (s)

Controller AGMMPFCMMPFC-PID PID 46.71 57.41 49.12 54.02 0.651 3.001 2 0.70 2.522

PID 81.01 75.32 1.401 1.352

Note. 1 means the sudden decrease of 100% load process, 2 means the sudden increase of 100% load process.

(|Δ𝑛𝑒 |) and integral absolute error (IAE), they are 5.4 rpm and 70.07, followed by the MMPFC-PID and PID. It indicates that by introducing the open-loop gain in the AGMMPFC-PID to modify the predictive multimodels the fluctuation of engine speed is reduced significantly. It also proves that, even without the model modifying in the MMPFC-PID controller, it still can keep stable and sustain better performance than in the well-tuned PID, but without the same complex parameters adjusting process. The speed responses of sudden decrease 100% load and sudden increase 100% load conditions are shown in Figures 16(b) and 16(c), respectively. The detailed performance indices are given in Table 6. From Figures 16(b) and 16(c) and Table 6, we know that the absolute engine speed variations in the original MMPFC-PID are around 40% less than those in the PID case, but recovery time values are longer. From Figure 17(c),

we observe that the model mismatch would be very serious under the mutation load conditions: the open-loop gain under the full load condition is about eight, but when without load it becomes around fifty; it means the process gain in the predictive multimodels would change more than six times. It indicates that the MMPFC-PID still has a good closedloop stability and can keep an acceptable performance when predictive multimodels deviate largely. Moreover, with the parameters updating in predictive multimodels by identifying the open-loop gain online, the performances of the AGMMPFC-PID in criteria of absolute engine speed variation and recovery time have been improved to be further better than those in the MMPFC-PID and PID during the sudden load changing processes. By analyzing the response curves of load (Figure 17(a)), control input (Figure 17(b)), and the identified open-loop gain (Figure 17(c)), it is easy to understand that the estimated open-loop gain can properly reflect the load changing in time. It can modify the predictive multimodels online; thus the parameters in PID controller can be optimized by the multimodel PFC method in each control step. As a result, the control input can be regulated faster to restrict the disturbances.

5. Conclusion To upgrade speed tracking accuracy and enhance disturbance suppression ability for the marine main engine, a novel method is proposed by combining the MMPC theory with PID algorithm, which reduces parameters regulation as well. In this method, the receding-horizon optimization process


17 2100

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Figure 16: Detailed engine speed responses in wave load (a), sudden decrease of 100% load (b), and sudden increase of 100% load (c).

is carried out based on the MMPFC scheme. To alleviate the model mismatch when load disturbances occur, the estimation of the open-loop gain is adopted to modify the predictive multimodels online. Meanwhile, a cycle-detailed hybrid nonlinear engine model is established to test the proposed controllers. The simulation experiments demonstrate the following: (1) The MMPFC-PID controller has more adaptive ability than the well-tuned PID in keeping better steady and transient performance under different propulsion load factors during the whole speed range. (2) The MMPFC-PID controller is capable of guaranteeing system stability when the disturbance loads cause the predictive model to deviate largely from the designed condition.

(3) Considering the wave disturbance, system performance under the MMPFC-PID controller is better than that in the well-tuned PID. However, the MMPFC-PID controller does not need difficult parameters adjusting process. Under the mutation load conditions, the MMPFC-PID has fewer speed variations but longer setting time compared with the well-tuned PID. (4) By estimating the open-loop gain and modifying the predictive multimodels online, the AGMMPFC-PID controller obtains better performance than the MMPFC-PID and well-tuned PID controllers in the sense of overcoming wave load and mutation load disturbances.

Conflicts of Interest The authors do not have any conflicts of interest.

18



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(c)

Figure 17: The demonstration of the load changing (a), control input changing (b), and identified open-loop gain changing (c) in disturbance loads.

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